Science's Breakthrough of the Year

johkir writes "Last year, evolution was the breakthrough of the year; We found it full of new developments in understanding how new species originate. But we did get a complaint or two that perhaps we were just paying extra attention to the lively political/religious debate that was taking place over the issue, particularly in the United States. Perish the thought! Our readers can relax this year: Religion and politics are off the table, and n-dimensional geometry is on instead. This year's Breakthrough salutes the work of a lone, publicity-shy Russian mathematician named Grigori Perelman, who was at the Steklov Institute of Mathematics of the Russian Academy of Sciences until 2005. The work is very technical but has received unusual public attention because Perelman appears to have proven the Poincaré Conjecture (Our coverage from earlier this year), a problem in topology whose solution will earn a $1 million prize from the Clay Mathematics Institute. That's only if Perelman survives what's left of a 2-year gauntlet of critical attack required by the Clay rules, but most mathematicians think he will. There is also a page of runner-ups. Many of which have been covered here on Slashdot."

Re:Update please

[Wooloomooloo]

He turned the prize down. In fact, he didn't even show up at the ceremony.

It's runners-up, not runner-ups.

[isaac]

In case you were sick that day in remedial English 101, noun-adjective compounds - attorney general, mother-in-law, runner-up - are made plural by pluralizing the noun: attorneys general, mothers-in-law, runners-up.

-Isaac

Re:Update please

[modular_form]

He turned down the Fields medal, but the million dollars is a separate thing. They won't even offer it until two years after his proof is published. I heard the man lives on $1 a day, so he's probably not interested in the money either.

There's a podcast as well

[ahab_2001]

This all comes from the 22 December issue of the journal Science, in case that wasn't clear from the original posting. All of the stories from the issue are indexed here [sciencemag.org]; to get access to the articles I believe you need to register with the site. There's also a podcast [sciencemag.org], which doesn't require registration.

Re:Religion and politics off the table? I think no

[c_forq]

I think you are a little off. The Apocrypha isn't a book, it is a collection of books, and a couple different versions of books existing in the canonical bible. At the time of Jesus the Apocryphal books were debated in the Jewish community, and in the modern world, besides a couple of extremely small Judaism sects, I believe only the Roman Catholic and Eastern Orthodox churches use it, but could be wrong. The reason the Apocrypha is not included in the normal canon of the bible is usually accredited to lacking authenticity, or conflicting with established books.

Pereleman isn't accepting for a reason.

[Starker_Kull]

He thinks that academia is littered with people who are more interested in promoting themselves than who are actually good at research, and this leads to a lot more politicing than researching, and the system is set up to promote that. This is the reason he is not interested in claiming prize money or prizes or other official recognition of his worth. I don't necessarily agree with that point of view, but perhaps it is worth considering if he has a legitimate gripe? There is a good article about him in the New Yorker Mag; here is the link and concluding paragraphs:

http://www.newyorker.com/fact/content/articles/060 828fa_fact2 [newyorker.com]

As for Yau, Perelman said, "I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest." The prospect of being awarded a Fields Medal had forced him to make a complete break with his profession. "As long as I was not conspicuous, I had a choice," Perelman explained. "Either to make some ugly thing"--a fuss about the math community's lack of integrity--"or, if I didn't do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit." We asked Perelman whether, by refusing the Fields and withdrawing from his profession, he was eliminating any possibility of influencing the discipline. "I am not a politician!" he replied, angrily. Perelman would not say whether his objection to awards extended to the Clay Institute's million-dollar prize. "I'm not going to decide whether to accept the prize until it is offered," he said. Mikhail Gromov, the Russian geometer, said that he understood Perelman's logic: "To do great work, you have to have a pure mind. You can think only about the mathematics. Everything else is human weakness. Accepting prizes is showing weakness." Others might view Perelman's refusal to accept a Fields as arrogant, Gromov said, but his principles are admirable. "The ideal scientist does science and cares about nothing else," he said. "He wants to live this ideal. Now, I don't think he really lives on this ideal plane. But he wants to."

The Article

[Starker_Kull]

The Poincare Conjecture-Proved: The solution of a century-old mathematics problem turns out to be a bittersweet prize

TO MATHEMATICIANS, GRIGORI PERELMAN'S proof of the Poincare conjecture qualifies at least as the Breakthrough of the Decade. But it has taken them a good part of that decade to convince themselves that it was for real. In 2006, nearly 4 years after the Russian mathematician released the first of three papers outlining the proof, researchers finally reached a consensus that Perelman had solved one of the subject's most venerable problems. But the solution touched off a storm of controversy and drama that threatened to overshadow the brilliant work.

Perelman's proof has fundamentally altered two distinct branches of mathematics. First, it solved a problem that for more than a century was the indigestible seed at the core of topology, the mathematical study of abstract shape. Most mathematicians expect that the work will lead to a much broader result, a proof of the geometrization conjecture: essentially, a "periodic table" that brings clarity to the study of three-dimensional spaces, much as Mendeleev's table did for chemistry.

While bringing new results to topology, Perelman's work brought new techniques to geometry. It cemented the central role of geometric evolution equations, powerful machinery for transforming hard-to-work-with spaces into more-manageable ones. Earlier studies of such equations always ran into "singularities" at which the equations break down. Perelman dynamited that roadblock.

"This is the first time that mathematicians have been able to understand the structure of singularities and the development of such a complicated system," said Shing-Tung Yau of Harvard University at a lecture in Beijing this summer. "The methods developed ... should shed light on many natural systems, such as the Navier-Stokes equation [of fluid dynamics] and the Einstein equation [of general relativity]."

Unruly spaces

Henri Poincare, who posed his problem in 1904, is generally regarded as the founded of topology, the first mathematician to clearly distinguish it from analysis (the branch of mathematics that evolved from calculus) and geometry. Topology is often described as "rubber-sheet geometry," because it deals with properties of surfaces that can undergo arbitrary amounts of stretching. Tearing and its opposite, sewing, are not allowed.

Our bodies, and most of the familiar objects they interact with, have three dimensions. Their surfaces, however, have only two. As far as topology is concerned, two-dimensional surfaces with no boundary (those that wrap around and close in on themselves, as our skin does) have essentially only one distinguishing feature: the number of holes in the surface. A surface with no holes is a sphere: a surface with one hole is a torus; and so on. A sphere can never be turned into a torus, or vice versa.

Three-dimensional objects with 2D surfaces, however, are just the beginning. For example, it is possible to define curved 3D spaces as boundaries of 4D objects. Human beings can only dimly visualize such spaces, but mathematicians can use symbolic notation to describe them and explore their properties. Poincare developed and ingenious tool called the "fundamental group," for detecting holes, twists, and other feature in spaces of any dimension. He conjectured that a 3D space cannot hide any interesting topology from the fundamental group. That is, a 3D space with a "trivial" fundamental group must be a hypersphere: the boundary of a ball in 4D space.

Although simple to state, Poincare's conjecture proved maddeningly difficult to prove. By the early 1980's, mathematicians had proved analogous statements for spaces of every dimension higher than three - but not for the original one that Poincare had pondered.

To make progress, topologists reached for a tool they had neglected: a way to specify distance. They se